March 3

Question 1

a) Compute
$\left(\frac{14}{101}\right)$

b) Compute
$\left(\frac{-32}{101}\right)$

Question 2

If
$p=2^k+1$
is prime, show that 3 is not a quadratic residue.

Combine this with the result of question 2 from March 1 to conclude that every number not divisible by p is congruent to a power of 3 modulo p.

Question 3

For what primes p the number 5 is a quadratic residue?

Question 4

Show that any prime number p which is congruent to 1 modulo 6 is a divisor of some number of the form
$n^2+3$